Question

if a and b are are (2,1)and 3,4 respectively find the coordinates of p such tha ap= 4/5 ab and p lies online sengment ab

Answer 1

Answer:

ꍌ꒐꒦ꏂꋊ:- ꋬ ꒒꒐ꋊꏂ ꇙꏂꍌꂵꏂꋊ꓄ ꒻ꄲ꒐ꋊ꒐ꋊꍌ ꓄ꁝꏂ ꉣꄲ꒐ꋊ꓄ꇙ ꋬ(−2,−2) ꋬꋊ꒯ ꃳ(2,−4). ꉣ ꒐ꇙ ꋬ ꉣꄲ꒐ꋊ꓄ ꄲꋊ ꋬꃳ ꇙ꒤ꉔꁝ ꓄ꁝꋬ꓄ ꋬꉣ=

7

3

ꋬꃳ.

ꋊꄲꅐ,

ꋬꉣ=

7

3

ꋬꃳ(ꍌ꒐꒦ꏂꋊ)

7ꋬꉣ=3(ꋬꉣ+ꃳꉣ)

7ꋬꉣ=3ꋬꉣ+3ꃳꉣ

⇒7ꋬꉣ−3ꋬꉣ=3ꃳꉣ

⇒

ꃳꉣ

ꋬꉣ

=

4

3

꓄ꁝꏂꋪꏂꊰꄲꋪꏂ,

ꉣꄲ꒐ꋊ꓄ ꉣ ꒯꒐꒦꒐꒯ꏂꇙ ꋬꃳ ꒐ꋊ꓄ꏂꋪꋊꋬ꒒꒒ꌦ ꒐ꋊ ꓄ꁝꏂ ꋪꋬ꓄꒐ꄲ 3:4.

ꋬꇙ ꅐꏂ ꀘꋊꄲꅐ ꓄ꁝꋬ꓄ ꒐ꊰ ꋬ ꉣꄲ꒐ꋊ꓄ (ꁝ,ꀘ) ꒯꒐꒦꒐꒯ꏂꇙ ꋬ ꒒꒐ꋊꏂ ꒻ꄲ꒐ꋊ꒐ꋊꍌ ꓄ꁝꏂ ꉣꄲ꒐ꋊ꓄ (ꉧ

1

,ꌦ

1

) ꋬꋊ꒯ (ꉧ

2

,ꌦ

2

) ꒐ꋊ ꓄ꁝꏂ ꋪꋬ꓄꒐ꄲꋊ ꂵ:ꋊ, ꓄ꁝꏂꋊ ꉔꄲꄲꋪ꒯꒐ꋊꋬ꓄ꏂꇙ ꄲꊰ ꓄ꁝꏂ ꉣꄲ꒐ꋊ꓄ ꒐ꇙ ꍌ꒐꒦ꏂꋊ ꋬꇙ-

(ꁝ,ꀘ)=(

ꂵ+ꋊ

ꂵꉧ

2

+ꋊꉧ

1

,

ꂵ+ꋊ

ꂵꌦ

2

+ꋊꌦ

1

)

꓄ꁝꏂꋪꏂꊰꄲꋪꏂ,

ꉔꄲꄲꋪ꒯꒐ꋊꋬ꓄ꏂꇙ ꄲꊰ ꉣ=(

3+4

3×(2)+4×(−2)

,

3+4

3×(−4)+4×(−2)

)=(

7

−2

,

7

−20

)

ꁝꏂꋊꉔꏂ, ꓄ꁝꏂ ꉔꄲꄲꋪ꒯꒐ꋊꋬ꓄ꏂꇙ ꄲꊰ ꉣ ꋬꋪꏂ (

7

−2

,

7

−20

)

Answer 2

$\huge{\underline{\bf\red{Questi {\mathbb{O}} n} :}}$

$\sf If \: A \: and \: B \: are \: (2,1) \: and \: (3,4) \: respectively, \\ \sf then \: find \: the \: coordinates \: of \: P \: such \: that \\ \sf AP=\dfrac{4}{5}AB \: and \: P \: lies \: on \: line \: segment \: AB.$

$\huge{\underline{\: \bf\green{Answ {\mathbb{E}} r}\ \: :}}$

• $\sf P \equiv \bigg( \dfrac{14}{5} , \dfrac{17}{5} \bigg)$

$\Large{\underline{\bf\purple{Giv {\mathbb{E}} n}\ :}}$

• $\sf A \equiv(2,1) \equiv(x_{1}, y_{1})$

• $\sf B \equiv(3,4)\equiv(x_{2}, y_{2})$

• $\sf \dfrac{AP}{AB} = \dfrac{4}{5}$

$\Large{\underline{\bf\purple{To \ Fi {\mathbb{N}} d}\ :}}$

• $\sf P = (x, y)$

$\huge{\underline{\bf\blue{Soluti {\mathbb{O}} n}\ :}}$

let the ratio b/w AP and AB be 4k : 5k.

Therefore, the ratio between AP and PB will be 4k : k

That is,

$\sf AP : AB = 4k : k$

Thus, the point P divides the line segment AB in the ratio 4 : 1.

Thus,

$\begin{aligned} \sf P & \equiv( x, y) \\ \\ & \equiv \bigg[ \frac{ 4x_2 + x_1}{4 + 1} , \frac{4y _2 +y_1}{4 + 1} \bigg] \\ \\ &\equiv \bigg[ \frac{ 4(3) + (2)}{5} , \frac{4(4) +(1)}{5} \bigg] \\ \\ &\equiv \bigg[ \frac{12 + 2}{5} , \frac{16 + 1}{5} \bigg] \\ \\ &\equiv \bigg[ \frac{14}{5} , \frac{17}{5} \bigg] \\ & & & & & \end{aligned}$

$\red{\rule{5.5cm}{0.02cm}}$

$\purple{\rule{7.5cm}{0.02cm}}$

$\Large{ \mid {\underline{\underline{\bf\green{BrainLiest \ AnswEr}}}} \mid }$